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Mathematics
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Mathematical Game Theory
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Mathematical Game Theory

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 August 2018) | Viewed by 27827

Special Issue Editor

Prof. Dr. Vladimir Mazalov

E-Mail Website
Guest Editor
Institute of Applied Mathematical Research of Karelian Research Centre, Russian Academy of Sciences, Petrozavodsk, 185910 Karelia, Russia
Interests: game theory; decision analysis; dynamic programming; bargaining models; networking games; behavioral models
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Rapid developments in technology, communication, industrial organization, economic integration and international trade have stimulated the appearance of different practical statements in the description of agent interaction, based on the game theory. A strategic approach to decision-making is very useful in many areas, such as bargaining, resource allocation, fishery, competition and cooperation, pollution control, networking, and competitive mobile systems. The main tools in the analysis of game models are mathematical methods. In dynamic games, the Hamilton-Jacobi-Bellman equation and Pontryagin maximum principle are very useful. Dynamic games theory has many applications in many fields, including biology, computer science, ecology, economics and management. In networking games, the result of interactions between agents are defined by a certain network. Networking games are games on graphs; graph-theoretic models are very important in this field.  This direction in game theory has appeared in connection with the emergence of new information technologies. First of all it's the global Internet, mobile communications, distributed and cloud computing and social networks. In routing games, players choose information transfer channels with limited bandwidths.  Equilibrium, here, is a result of the application of the optimization theory.  Social networks appear lead to many new game-theoretic problem formulations. Users of such networks are united in communities, forming networks of different topologies. An analysis of the structure of such a graph is important in of itself, but is also important in being able to evaluate the results of equilibrium game-theoretic interactions in such networks. The spectrum of mathematical approaches in game theory is very wide.

This Special Issue contains papers that cover the wide range of mathematical methods used in game theory, including recent advances in areas of high potential for future works, as well as new developments in classical results. It will be of interest to anyone doing theoretical research in game theory or working on one its numerous applications.

Prof. Dr. Vladimir Mazalov
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Competition and cooperation
  • Dynamic games
  • Networking games
  • Behavioral game theory
  • Potential games
  • Bargaining models
  • Hamilton-Jacobi-Bellman equation
  • Pontryagin maximum principle
  • Applications in resource allocation, fishery, pollution control, networking

Published Papers (8 papers)

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Research

19 pages, 392 KiB  
Article
Price and Treatment Decisions in Epidemics: A Differential Game Approach
by Andrea Di Liddo
Mathematics 2018, 6(10), 190; https://doi.org/10.3390/math6100190 - 02 Oct 2018
Cited by 2 | Viewed by 2134
Abstract
We consider a pharmaceutical company that sells a drug that is useful in the treatment of an infectious disease. A public authority buys the drug to heal at least a portion of the infected population. The authority has an overall budget for all [...] Read more.
We consider a pharmaceutical company that sells a drug that is useful in the treatment of an infectious disease. A public authority buys the drug to heal at least a portion of the infected population. The authority has an overall budget for all health care costs in the country and can only allocate a (small) part of the budget to the purchase of the drug. The government chooses the amount of drug to be purchased in order to minimize both the number of infectious people and the perceived cost of the operation along a given time horizon. This cost can be modeled through a linear or quadratic function of the monetary cost (as generally happens in the literature) or through a specific function (blow-up) that makes the budget constraint endogenous. The pharmaceutical company chooses the price of the drug in order to maximize its profit and knowing the budget constraints of the buyer. The resulting differential game is studied by supposing the simplest possible dynamics for the population. Two different games are proposed and their solutions are discussed: a cooperative game in which the two players bargain for the price of the drug and the quantity is purchased with the aim of maximizing the overall payoff and a competitive game in which the seller announces a price strategy to the buyer and binds to it; the buyer reacts by choosing the quantity to be purchased. In the case of linear or quadratic costs, the solution provided (for budget levels is not high enough) that the government spends the entire budget to purchase the drug. This drawback does not occur when the blow-up cost function is used. Full article
(This article belongs to the Special Issue Mathematical Game Theory)
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17 pages, 363 KiB  
Article
Payoff Distribution in a Multi-Company Extraction Game with Uncertain Duration
by Ekaterina Gromova, Anastasiya Malakhova and Arsen Palestini
Mathematics 2018, 6(9), 165; https://doi.org/10.3390/math6090165 - 11 Sep 2018
Cited by 8 | Viewed by 2935
Abstract
A nonrenewable resource extraction game model is analyzed in a differential game theory framework with random duration. If the cumulative distribution function (c.d.f.) of the final time is discontinuous, the related subgames are differentiated based on the position of the initial instant with [...] Read more.
A nonrenewable resource extraction game model is analyzed in a differential game theory framework with random duration. If the cumulative distribution function (c.d.f.) of the final time is discontinuous, the related subgames are differentiated based on the position of the initial instant with respect to the jump. We investigate properties of optimal trajectories and of imputation distribution procedures if the game is played cooperatively. Full article
(This article belongs to the Special Issue Mathematical Game Theory)
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14 pages, 2624 KiB  
Article
A Game-Theoretic Loss Allocation Approach in Power Distribution Systems with High Penetration of Distributed Generations
by Farzaneh Pourahmadi and Payman Dehghanian
Mathematics 2018, 6(9), 158; https://doi.org/10.3390/math6090158 - 06 Sep 2018
Cited by 8 | Viewed by 3446
Abstract
Allocation of the power losses to distributed generators and consumers has been a challenging concern for decades in restructured power systems. This paper proposes a promising approach for loss allocation in power distribution systems based on a cooperative concept of game-theory, named Shapley [...] Read more.
Allocation of the power losses to distributed generators and consumers has been a challenging concern for decades in restructured power systems. This paper proposes a promising approach for loss allocation in power distribution systems based on a cooperative concept of game-theory, named Shapley Value allocation. The proposed solution is a generic approach, applicable to both radial and meshed distribution systems as well as those with high penetration of renewables and DG units. With several different methods for distribution system loss allocation, the suggested method has been shown to be a straight-forward and efficient criterion for performance comparisons. The suggested loss allocation approach is numerically investigated, the results of which are presented for two distribution systems and its performance is compared with those obtained by other methodologies. Full article
(This article belongs to the Special Issue Mathematical Game Theory)
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9 pages, 266 KiB  
Article
Dynamic Multicriteria Games with Finite Horizon
by Anna Rettieva
Mathematics 2018, 6(9), 156; https://doi.org/10.3390/math6090156 - 05 Sep 2018
Cited by 16 | Viewed by 2551
Abstract
The approaches to construct optimal behavior in dynamic multicriteria games with finite horizon are presented. To obtain a multicriteria Nash equilibrium, the bargaining construction (Nash product) is adopted. To construct a multicriteria cooperative equilibrium, a Nash bargaining scheme is applied. Dynamic multicriteria bioresource [...] Read more.
The approaches to construct optimal behavior in dynamic multicriteria games with finite horizon are presented. To obtain a multicriteria Nash equilibrium, the bargaining construction (Nash product) is adopted. To construct a multicriteria cooperative equilibrium, a Nash bargaining scheme is applied. Dynamic multicriteria bioresource management problem with finite harvesting times is considered. The players’ strategies and the payoffs are obtained under cooperative and noncooperative behavior. Full article
(This article belongs to the Special Issue Mathematical Game Theory)
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9 pages, 462 KiB  
Article
Accounting Games: Using Matrix Algebra in Creating the Accounting Models
by Anna Vysotskaya
Mathematics 2018, 6(9), 152; https://doi.org/10.3390/math6090152 - 31 Aug 2018
Viewed by 4741
Abstract
The aim of this paper is to show the mathematical basis for a precise treatment of double-entry bookkeeping, which was first developed in the nineteenth century by Sir William Rowan Hamilton. This is done by using basic notions of matrix algebra founded on [...] Read more.
The aim of this paper is to show the mathematical basis for a precise treatment of double-entry bookkeeping, which was first developed in the nineteenth century by Sir William Rowan Hamilton. This is done by using basic notions of matrix algebra founded on the idea of ordered pairs. We also reveal how complex numbers and rationals (fractions) developed in mainstream accounting science and became a leading platform for the ongoing processes within Industry 4.0. The paper concludes with examples of how accounting operations can be represented by matrix equations with the result of generating a final report. The author presents a mathematical model of accounting which is independent of specific existential forms, but which is capable of undertaking the form of any of them and thus which has the potential of being understood and accepted by specialists globally. Full article
(This article belongs to the Special Issue Mathematical Game Theory)
9 pages, 239 KiB  
Article
Inverse Stackelberg Solutions for Games with Many Followers
by Yurii Averboukh
Mathematics 2018, 6(9), 151; https://doi.org/10.3390/math6090151 - 30 Aug 2018
Cited by 1 | Viewed by 2301
Abstract
The paper is devoted to inverse Stackelberg games with many players. We consider both static and differential games. The main assumption of the paper is the compactness of the strategy sets. We obtain the characterization of inverse Stackelberg solutions and under additional concavity [...] Read more.
The paper is devoted to inverse Stackelberg games with many players. We consider both static and differential games. The main assumption of the paper is the compactness of the strategy sets. We obtain the characterization of inverse Stackelberg solutions and under additional concavity conditions, establish the existence theorem. Full article
(This article belongs to the Special Issue Mathematical Game Theory)
10 pages, 237 KiB  
Article
A Markovian Mechanism of Proportional Resource Allocation in the Incentive Model as a Dynamic Stochastic Inverse Stackelberg Game
by Grigory Belyavsky, Natalya Danilova and Guennady Ougolnitsky
Mathematics 2018, 6(8), 131; https://doi.org/10.3390/math6080131 - 30 Jul 2018
Cited by 4 | Viewed by 2770
Abstract
This paper considers resource allocation among producers (agents) in the case where the Principal knows nothing about their cost functions while the agents have Markovian awareness about his/her strategies. We use a dynamic setup of the stochastic inverse Stackelberg game as the model. [...] Read more.
This paper considers resource allocation among producers (agents) in the case where the Principal knows nothing about their cost functions while the agents have Markovian awareness about his/her strategies. We use a dynamic setup of the stochastic inverse Stackelberg game as the model. We suggest an algorithm for solving this game based on Q-learning. The associated Bellman equations contain functions of one variable for the Principal and also for the agents. The new results are illustrated by numerical examples. Full article
(This article belongs to the Special Issue Mathematical Game Theory)
26 pages, 384 KiB  
Article
Finite Automata Capturing Winning Sequences for All Possible Variants of the PQ Penny Flip Game
by Theodore Andronikos, Alla Sirokofskich, Kalliopi Kastampolidou, Magdalini Varvouzou, Konstantinos Giannakis and Alexander Singh
Mathematics 2018, 6(2), 20; https://doi.org/10.3390/math6020020 - 01 Feb 2018
Cited by 16 | Viewed by 5122
Abstract
The meticulous study of finite automata has produced many important and useful results. Automata are simple yet efficient finite state machines that can be utilized in a plethora of situations. It comes, therefore, as no surprise that they have been used in classic [...] Read more.
The meticulous study of finite automata has produced many important and useful results. Automata are simple yet efficient finite state machines that can be utilized in a plethora of situations. It comes, therefore, as no surprise that they have been used in classic game theory in order to model players and their actions. Game theory has recently been influenced by ideas from the field of quantum computation. As a result, quantum versions of classic games have already been introduced and studied. The P Q penny flip game is a famous quantum game introduced by Meyer in 1999. In this paper, we investigate all possible finite games that can be played between the two players Q and Picard of the original P Q game. For this purpose, we establish a rigorous connection between finite automata and the P Q game along with all its possible variations. Starting from the automaton that corresponds to the original game, we construct more elaborate automata for certain extensions of the game, before finally presenting a semiautomaton that captures the intrinsic behavior of all possible variants of the P Q game. What this means is that, from the semiautomaton in question, by setting appropriate initial and accepting states, one can construct deterministic automata able to capture every possible finite game that can be played between the two players Q and Picard. Moreover, we introduce the new concepts of a winning automaton and complete automaton for either player. Full article
(This article belongs to the Special Issue Mathematical Game Theory)
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